The Independent article that you linked has most of the details right. The key phrase is this:

Spending during the NHS’s first year vastly overshot the budget

This raises an important question: what was the budget? Knowing the answer to this question will, I think, answer your first question- was it underfunded, or overly expensive?

Indeed, NHS historian Charles Webster has noted that the early NHS—then as now—always did battle with the pervasive assumption that spending was off the charts and that the NHS was a financial sinkhole. Because of this perception, in 1950 expenditure for the NHS was tightly restrained. Yet the Guillebaud committee of 1956 found that actually the NHS was pretty effectively administered.

Building off Webster's insights, historian Tony Cutler argues that the real issue lies in the budget itself, and the financial estimates of the NHS made in two separate government documents, one dating to 1944 and another to 1946. These estimates had erroneous assumptions about the NHS and its functionality hardwired into them. They were flawed, and all too frequently the numbers had been lowballed. But these estimates floated in government white papers were taken not as estimates to be adjusted as necessary, but as hard proclamations of how much the NHS should cost. Therefore, when spending “overshot the budget,” the government looked to revise the NHS rather than the budget itself.

So why were the estimates flawed, and why were they so far off? Here is what we have to remember: every single aspect of what the NHS would cover had to be worked out and then accounted for. The Beveridge Report of 1942, which laid out the basic ideas of the health service, was actually pretty vague when it came down to brass tacks; it left civil servants with quite a lot of work to do. The idea of Labour and Conservative coming together to sign off on a universal health service obscures the complicated process of planning, delineating, and decision making necessary to turn the Beveridge Report into reality. OK, so the government will take over hospitals—which ones? All of them? Or will it leave the voluntary, donation financed hospitals alone? Also, will doing one or the other of these options majorly upset the medical lobby, and what happens if they’re not happy with the NHS? It should have a dental service—who is covered under that? What about kids? Do they get their teeth checked under the NHS, or is that still under the auspices of the school medical service? Even the idea that the NHS to be free at the point of use had to be worked out (very early plans for the NHS dental service had people paying for their dentures) with implications for the cost projections.

Cutler’s point here is that as new ideas and permutations of the NHS were discussed, debated, and then set down as policy, the financial expenditure estimates never appropriately caught up with them. As the planned remit of the NHS expanded, the estimate of the cost didn’t expand with it. Nor did the planners account for the popularity of the NHS and the backlog of medical treatment that, with the advent of universal health care, was begging to be addressed. Let’s go back to the dental and opthalmic services, which is a key example that Cutler employs (along with pharmacies and hospitals). In 1944, the projected cost of D&O services was 10 million pounds per year for dental services, and 1 million pounds per year for opthalmic services. At the end of the NHS’ first fiscal year, dental services cost 29 million pounds, and opthalmic services 19.9 million. Yikes.

But let’s go back a little further. The first estimate of the dental service was made in 1943, and it suggested that the dental service would likely cost 30 million pounds per year, with 20 million of that coming from public funds. This estimate was based on the idea that 1) kids would get their service through the School Medical Service and local authorities/ county councils (so they would pick up some of the costs); 2) people would pay for their dentures, thus easing the financial costs, and 3) that 10 to 20 percent of adults would use the service every year, but all adults ages 15+ would be eligible for it. Then in 1944, fears that there wouldn’t be enough dentists to meet demand actually prompted civil servants to cut back their plans for the dental service. Maybe instead of having a really big dental service, the thought went, we’ll just invest more in children’s and family dental services and improve the dental health of the population from there. So having made this cutback, they cut the estimate accordingly, to 10 million.

And then the idea changed again, because another group of NHS planners (there were many planning committees) decided that actually they wanted to make dental services free at the point of use (no paying for dentures here) as with everything else. Then, while they were at it, they also that the whole idea of just focusing on children’s services was kind of bunk, because if you're going to the trouble of having a comprehensive medical service you ought to have a comprehensive dental service to go with it, etc. So going with this vastly expanded and free service now meant that that the 10 million figure needed further revision to account for the fact that everyone, and not just mothers and children, would be encouraged to use it, and the government wouldn't recoup any costs from dentures or glasses, etc. Except that didn’t happen. The 10 million figure stayed, so when the dental costs actually turned out to be 30 million pounds, it looked like spending was dangerously out of control. It wasn’t; the yardstick was wrong.

The other point that Cutler makes is that the NHS planners didn’t always account for demand of services. One of the big things that the NHS did was begin to redress very long standing inequities in the provision and quality of health services across the country—to make it possible for a poor farmer in rural Shropshire to get the same care as a wealthy businessman in west London. But the estimates made little allowance for the initial expenditures and investments that such a project would need. In the case of opthalmic services, for example, the plans just took the existing amount of people receiving benefits for glasses under the old insurance system, and scaled it up. It didn’t account for the fact that there were lots of people who didn’t bother getting glasses from an ophthalmologist or optician, because it was expensive, and wore cheap glasses from the department store—but who would jump at an opportunity to get new ones if they were covered. So demand, and cost, spiked. This was not accounted for, and this happened not just with glasses but also with dentures and prescriptions (and in a more complicated way) with hospital services.

As the Independent notes, the Guillebaud Committee's findings—that the NHS was actually cost-effective— did lead to less tightening of the belts, and more spending on and growth of the NHS. Some historians, such as Charles Webster, have challenged the sanguine idea of Tory consensus and instead highlighted more retrenchment and reluctance, so the Independent might be putting too neat a bow on it. (I would try to give you a proper answer to this part of question, but once again I have been felled by the fact that I am traveling and do not have much of what I would normally reference to hand.) I hope this helps explain the idea of the NHS having “overshot the budget” and the early politics of NHS financing. Once early estimates are out there, no matter how flawed they may be, they can become (as Cutler puts it) “questionable standards by which parsimony [is] regularly justified.”

Sources:

• Charles Webster, The National Health Service: A Political History (Oxford, 1998)

• Tony Cutler, “Dangerous Yardsticks? Early Cost Estimates and the Politics of Financial Management in the First Decade of the National Health Service” Medical History 47 (2003): 217-238

• Jim Tomlinson, “Welfare and the Economy: The Economic Impact of the Welfare State, 1945-51” Twentieth Century British History 6 (1995): 194-219.